I have an interpolating sequence for $H^\infty(\mathbb D)$ in the unit disk (that is, a uniformly separated sequence) which, in addition, satisfies the Blaschke condition. Is it possible that each point on the unit circle $\mathbb T$ is an accumulation point of the sequence?
For me this is hard to believe. At least, if each (or almost every) of these accumulation points is non-tangential, then the limit function of the corresponding Blaschke product is zero, hence the Blaschke product itself must be zero. Contradiction! I know that this is not a proof (otherwise, I would not ask here). Moreover, this "argument" does not use the uniform separation of the sequence. Does anyone know more?